The Purdue Rare Isotope Measurement Laboratory (PRIME Lab) is a dedicated research and service facility for accelerator mass spectrometry (AMS). AMS is an ultra-sensitive analytical technique for measuring low levels of long-lived radionuclides and rare trace elements. At Purdue we have constructed a major national AMS facility centered around the Physics Department's tandem electrostatic accelerator. We are using the accelerator to measure both man-made and cosmic-ray-produced radionuclides such as 10Be (half-life 1.36 +/- 0.07 My [Nishiizumi et al. 2007]), 14C (5730 years), and 36Cl (300,000 years) in natural samples having isotopic abundances down to one part in 1x1015 (a thousand million million).